is a viscosity solution of \(L[Z] \ge 1\)  (since the maximum of two viscosity sub-solutions is a viscosity sub-solution [1]} and \(L[Z]=+\infty \)  on \(\partial B_R(0)\) ). Also notice that \(\partial _t \varphi = 0\)  for \(|x| \ge R\) , and \(|\partial _t \varphi |=|\partial _t \phi | \le C\)  in \(B_R(0)\) , so \(\partial _t \varphi \)  is bounded.
